In this paper, we propose MUSE-RNN, a multilayer self-evolving recurrent neural network model for real-time classification of streaming data. Unlike the existing approaches, MUSE-RNN offers special treatment towards capturing temporal aspects of data stream through its novel recurrent learning approach based on the teacher forcing policy. Novelties here are twofold. First, in contrast to the traditional RNN models, MUSE-RNN has intrinsic ability to self-adjust its capacity by growing and pruning hidden nodes as well as layers, to handle the ever-changing characteristics of data stream. Second, MUSE-RNN adopts a unique scoring-based layer adaptation mechanism, which makes it capable of recalling prior tasks, with minimum exploitation of network parameters. The performance of MUSE-RNN is evaluated in comparison with a number of state-of-the-art techniques, using seven popular data streams and continual learning problems under prequential test-then-train protocol. Experimental results demonstrate the effectiveness of MUSE-RNN in stream classification scenario.